MinT - Best Known (96−41, 96, s)-Nets in Base 27

Best Known (96−41, 96, s)-Nets in Base 27

(96−41, 96, 248)-Net over F₂₇ — Constructive and digital

Digital (55, 96, 248)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 20, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (7, 27, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (8, 49, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(96−41, 96, 398)-Net in Base 27 — Constructive

(55, 96, 398)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (0, 20, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]
2. (35, 76, 370)-net in base 27, using
  - base change [i] based on digital (16, 57, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(96−41, 96, 1672)-Net over F₂₇ — Digital

Digital (55, 96, 1672)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(96−41, 96, 2010520)-Net in Base 27 — Upper bound on s

There is no (55, 96, 2010521)-net in base 27, because

1 times m-reduction [i] would yield (55, 95, 2010521)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 9540 232999 381750 808129 227210 337503 336050 210512 038890 150416 675552 068427 362579 223927 842377 923648 190524 408860 794146 425143 108654 863749 241841 > 27⁹⁵ [i]